Let T be linear transformation of $R^{3}$ into $R^{2}$ defined by T( x, y, z ) = (2x + y - z, 3x- 2 y + 4z) for all ( x, y, z ) in $R^{3}$. Then
the matrix of T relative to the bases $\beta=\left\{\epsilon_{1}=(1,1,1),\epsilon_{2}=(1,1,0),\epsilon_{3}=(1,0,0)\right\}$ and $\delta=\left\{\eta_{1}=(1,3),\eta_{2}=(1,4) \right\}$ is:
1). $\begin{bmatrix}3 & 11 & 5 \\-1 & -8 & -3 \end{bmatrix}$
2). $\begin{bmatrix}3 & 11 & -5 \\1 & -8 & 3 \end{bmatrix}$
3). $\begin{bmatrix}-3 & 11 & 5 \\1 & 8 & 3 \end{bmatrix}$
4). $\begin{bmatrix}3 & -11 & 5 \\-1 & 8 & 3 \end{bmatrix}$
When the mode is ill-defined, then the Empirical mode formula is of the form:
1). 3 Median-Mode = 2 Mean
2). Mean-Mode = 2[Mean-Median]
3). Mean-Mode = 3[Median-Mean]
4). Mean-Median = 3[Mean - Mode]
The general solution of the equation $\frac{dy}{dx}=(\frac{y}{x})+\phi(\frac{x}{y})$ is given by y ln lCxl = x. Then $\phi(\frac{x}{y})$ is :
1). $\left(-\frac{x}{y}\right)^{2}$
2). $\left(\frac{y}{x}\right)^{2}$
3). $\left(-\frac{y}{x}\right)^{2}$
4). $\left(\frac{x}{y}\right)^{2}$
If the Laplace transform of a function $f(t)\ is \frac{1}{s(s+a)}$ then $f(t)$ is equal to :
1). $\frac{1}{a}(1-e^{-at}) $
2). $(1-e^{-at}) $
3). $\frac{1}{a}e^{-at}$
4). $ae^{-at}$
The number of ways to distribute 20 identical balls in 4 different boxes such that no box remains empty is:
1). 696
2). 323
3). 969
4). 52